Going further to the basic approximation involved in the TDTF, we shall present first some features of the original, stationary, Thomas Fermi (TF) theory.
In principle the approximation is derived considering an infinite system of non-interacting fermions. From the associated wave-functions, which are plane waves, the local density of kinetic energy is constructed and connected with the density of probability. This local form is used further in complex system with interactions to approximate also locally the true kinetic energy density. Besides the proper expression, it is useful to see how TF emerge as a genuine Density Functional Theory. It is known from Hohenberg-Kohn (H-K) theorems [1]} that given a system of particles with a two-body interaction \(U(\vec{r},\vec{r}^{\prime })\)  placed in an external potential \(V(\vec{r})\) , there is an unique functional of density called energy \(E[\rho ]\)  which is minimized only by the true ground state density of the system \(\rho _0(\vec{r})\) :
\(E[\rho ]=G[\rho (\vec{r})]+\int _{\mathbb {R}^3} \rho (\vec{r})[V(\vec{r})+\int _{\mathbb {R}^3}U(\vec{r},\vec{r}^{\prime })\rho (\vec{r}^{\prime })dr^{\prime 3}]dr^3\) 
